GK 2018 Q8
8. Square \(ABCD\) has sides of length 3 cm. The points \(M\) and \(N\) lie on \(AD\) and \(AB\) so that \(CM\) and \(CN\) split the square into three pieces of the same area.
What is the length of \(DM\)?
- A 0.5 cm
- B 1 cm
- C 1.5 cm
- D 2 cm
- E 2.5 cm
Grey Kangaroo 2018, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 2 cm
The area of square \(ABCD\) is (3 × 3) cm2 = 9 cm2. Hence the area of each piece is \(\tfrac{1}{3} \times 9\) cm2 = 3 cm2. Since the area of a triangle is equal to half its base multiplied by its perpendicular height, we have \(\tfrac{1}{2} \times DM \times DC\) = 3 cm2. Therefore \(\tfrac{1}{2} \times DM \times 3\) cm = 3 cm2 and hence the length of \(DM\) is 2 cm.
Solution © UK Mathematics Trust, from the GK 2018 solutions.